Question

find the value of k if 2x raised to the power 4 + 3 x cube + 2 k x square + 3 X + 6 is exactly divisible by X + 2​

Answer 1

ATQ, p(x) = 2x⁴ + 3x³ + 2kx² + 3x + 6 is exactly divisible by g(x) = x + 2

hence, remainder must come 0 when p(x) is divided by g(x)

let's find the value of x first.

➡ x + 2 = 0

➡ x = 0 - 2

➡ x = -2

putting value of x in p(x)

➡ p(-2) = 2(-2)⁴ + 3(-2)³ + 2k(-2)² + 3(-2) + 6

= 2(16) + 3(-8) + 2k(4) - 6 + 6

= 32 - 24 + 8k

= 8 + 8k --------(i)

since remainder is 0, let us equate (i) with 0

➡ 8 + 8k = 0

➡ 8k = -8

➡ k = -8/8

➡ k = -1

hence, the value of k is -1

Answer 2

$\huge\underline\textbf{Answer:-}$

• In ATQ p(x)

= 2x⁴ + 3x³ + 2kx² + 3x + 6 will be divisible by g(x) = x + 2

$\textbf\huge{ We need to get the reminder 0}$

$\textbf\huge{ when p(x) divided by g(x):-}$

$\huge\underline\textbf{Value-of(x)}$

$=> x + 2 = 0$

$=> x = 0 - 2$

$=> x = -2$

$\textbf{Adding the value x from p(x):-}$

=>p(-2)

=>2(-2)⁴ + 3(-2)³ + 2k(-2)² + 3(-2) + 6

$=>2(16) + 3(-8) + 2k(4) - 6 + 6$

$=>32 - 24 + 8k$

$=>8 + 8k (This-is-equation-(1)$

$\tiny\textbf{The reminder we get 0 so let us add with 0 and equation(1):-}$

$=> 8 + 8k = 0$

$=> 8k = -8$

$=> k = -8/8$

$=> k = -1$

$\textbf{Therefore,the value of k is -1}$